It follows from Corollory \(\ref {prethmcorollory}\)  that
\(\Phi _{E_i}\times \chi _t(\Gamma , E_i)=1\Leftrightarrow \mu _{E_i}^{alg}=0\text{, and }\lambda _{E_i,\Sigma _0}^{alg}=g.\) 
It is a standard result that if \(\mu _{E_1}^{alg}=0\)  then \(\mu _{E_2}^{alg}=0\) . On the other hand, it is shown by Greenberg and Vatsal that if
\(\mu _{E_1}^{alg}=\mu _{E_2}^{alg}=0\text{, then }\lambda _{E_1,\Sigma _0}^{alg}=\lambda _{E_2,\Sigma _0}^{alg}.\) 
This follows from [1]} and the discussion on [1]} preceding Remark (2.10). Therefore, it follows that
\(\Phi _{E_1}\times \chi _t(\Gamma , E_1)=1\Leftrightarrow \Phi _{E_2}\times \chi _t(\Gamma ,E_2)=1.\) 
By Lemma \(\ref {hypLemma}\) , if condition \((\star )\)  is satisfied then \(\Phi _{E_1}=\Phi _{E_2}=1\) . As a result, if in addition to conditions \((\ref {one})\)  to \((\ref {six})\) , condition \((\star )\)  is satisfied, then
\(\chi _t(\Gamma , E_1)=1\Leftrightarrow \chi _t(\Gamma ,E_2)=1.\) 
